cp's OEIS Frontend

A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.

%I A181821 #30 Feb 16 2025 08:33:13
%S A181821 1,2,4,6,8,12,16,30,36,24,32,60,64,48,72,210,128,180,256,120,144,96,
%T A181821 512,420,216,192,900,240,1024,360,2048,2310,288,384,432,1260,4096,768,
%U A181821 576,840,8192,720,16384,480,1800,1536,32768,4620,1296,1080,1152,960,65536
%N A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.
%C A181821 A permutation of A025487. a(n) is the member m of A025487 such that A181819(m) = n. a(n) is also the member of A025487 whose prime signature is conjugate to the prime signature of A108951(n).
%C A181821 If n = Product_i prime(e(i)) with the e(i) weakly decreasing, then a(n) = Product_i prime(i)^e(i). For example, 90 = prime(3) * prime(2) * prime(2) * prime(1), so a(90) = prime(1)^3 * prime(2)^2 * prime(3)^2 * prime(4)^1 = 12600. - _Gus Wiseman_, Jan 02 2019
%H A181821 Alois P. Heinz, <a href="/A181821/b181821.txt">Table of n, a(n) for n = 1..10000</a>
%H A181821 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConjugatePartition.html">Conjugate Partition </a>
%F A181821 If A108951(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A108951(n))).
%F A181821 a(A181819(n)) = A046523(n). - [See also A124859]. _Antti Karttunen_, Dec 10 2018
%F A181821 a(n) = A025487(A361808(n)). - _Pontus von Brömssen_, Mar 25 2023
%F A181821 a(n) = A108951(A122111(n)). - _Antti Karttunen_, Sep 15 2023
%e A181821 The canonical factorization of 24 is 2^3*3^1. Therefore, p(e(i)) = prime(3)*prime(1)(i.e., A000040(3)*A000040(1)), which equals 5*2 = 10. Since 24 is the smallest integer for which p(e(i)) = 10, a(10) = 24.
%p A181821 a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(sort(map(i->
%p A181821              numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
%p A181821 seq(a(n), n=1..70);  # _Alois P. Heinz_, Sep 05 2018
%t A181821 With[{s = Array[If[# == 1, 1, Times @@ Map[Prime@ Last@ # &, FactorInteger@ #]] &, 2^16]}, Array[First@ FirstPosition[s, #] &, LengthWhile[Differences@ Union@ s, # == 1 &]]] (* _Michael De Vlieger_, Dec 17 2018 *)
%t A181821 Table[Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,30}] (* _Gus Wiseman_, Jan 02 2019 *)
%o A181821 (PARI) A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); }; \\ _Antti Karttunen_, Dec 10 2018
%o A181821 (Python)
%o A181821 from math import prod
%o A181821 from sympy import prime, primepi, factorint
%o A181821 def A181821(n): return prod(prime(i)**e for i, e in enumerate(sorted(map(primepi,factorint(n,multiple=True)),reverse=True),1)) # _Chai Wah Wu_, Sep 15 2023
%Y A181821 Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181822.
%Y A181821 Cf. A046523, A181819, A181820.
%Y A181821 Cf. A000040, A001221, A001222, A002110, A056239, A071625, A112798, A118914, A122111, A124859, A182850, A305936, A361808.
%K A181821 nonn
%O A181821 1,2
%A A181821 _Matthew Vandermast_, Dec 07 2010
%E A181821 Definition corrected by _Gus Wiseman_, Jan 02 2019